A silent chain transmission mechanism, used to operate intake and exhaust valves, as well as a balancer or other mechanism, from the crankshaft of a four-cycle engine is shown in FIG. 4. The transmission includes a valve timing transmission mechanism M1 comprising a valve-driving crankshaft sprocket A11a, a camshaft sprocket A12a for the intake valves, a camshaft sprocket A12b for the exhaust valves, an endless, valve timing, silent chain 20a, in engagement with the sprockets, and a tensioner 30a, which maintains tension in the chain 20a. The transmission also includes a balancer transmission mechanism M2, comprising a balancer driving crankshaft sprocket A11b concentrically supported on the crankshaft with the valve-driving crankshaft sprocket A11a, a balancer shaft sprocket A13, an endless balancer driving silent chain 20b in engagement with sprockets A11b and A13, and a tensioner 30b, which maintains tension in chain 20b. 
The valve timing silent chain 20a and the balancer driving silent chain 20b, are both made up of interleaved link rows, each composed of a plurality of link plates arranged in parallel relationship to one another. Each link plate is formed by punching, and has a pair of teeth for meshing with the sprockets. The link plates in each row are gathered at random, and the rows are connected to one another by connecting pins.
When the valve-driving crankshaft sprocket A11a rotates through two revolutions, the intake valve camshaft and the exhaust valve camshaft each rotate through one revolution. Thus, a two to one speed reduction is required, and accordingly, the number of teeth on each of the camshaft sprockets A12a and A12b is twice the number of teeth of the valve-driving crankshaft sprocket A11a. 
When the crankshaft is rotates through one revolution, the balancer shaft rotates through two revolutions to compensate for the high degree of unbalance in the crankshaft. Thus, the rotational speed of the balancer shaft sprocket A13 must be twice that of the balancer driving crankshaft sprocket A11b, and accordingly, the number of teeth on the balancer driving crankshaft sprocket A11b must be twice the number of teeth on the balancer shaft sprocket A13.
Involute teeth, formed on the valve driving crankshaft sprocket A11a, the camshaft sprocket A12, and the balancer shaft sprocket A13 are all produced under the same tooth cutting conditions, by a hub cutter having a hob pitch which is the same as the chain pitch of the above-mentioned silent chains.
As shown in FIG. 5(a), even when the link plates in the valve timing chain 20a and the balancer driving chain 20b are punched accurately, slight shifts are generated in the positions of the connecting pin holes due to vibration during the punching operation. The positions of the pin holes in the link plate are not always symmetrical. Thus, the distance A between the one of the holes and the adjacent outer flank is different from the distance B between the other hole and the other outer flank. Similarly, the distance C between the center of one of the pin holes and a centerline is different from the distance D between the center of the other pin hole and the centerline. In other words, the relationships A=B, and C=D, as depicted in FIG. 5(c) are not always achieved. For example, the shift in the positions of the pin holes may be such that A>B and C<D, as shown in FIG. 5(a). When a large number of randomly selected link plates 21 are gathered in parallel in the direction of the chain width to form a link row during assembly of the chain, the outside dimensions, W and H, of the link rows, which are projections of all the parallel link plates in a widthwise row, are liable to be slightly larger than the outside dimensions, w and h, of an individual link plate, as shown in FIG. 5(b).
Accordingly, meshing failure can occur even where the chain is engaged with a sprocket designed for optimum engagement and produced using a hob cutter having a hob pitch the same as the chain pitch measured between a pair of pin holes punched in the link plates of the chain. Meshing failure may occur due to tight engagement between the sprocket and the link plates, as a result of the differences, δw and δh, between the outside dimensions W and H of a link row and the outside dimensions, w and h of an individual link plate, as shown in FIG. 5(b). The differences between the dimensions W and H on the one hand, and the dimensions w and h on the other, leads to problems of noise and vibration in the chain, and also to meshing interference at the meshing surfaces. Thus, the dimensional differences decrease the resistance of the silent chain to wear.
Furthermore, in a conventional silent chain transmission mechanism as shown in FIG. 4, all involute teeth of the various sprockets are produced using hob-cutters having the same hob pitch, even though the sprockets, except for the two camshaft sprockets, have different numbers of teeth. It is very difficult to attain optimum meshing between the chain and these sprockets across the board, and impossible to avoid meshing failure, which can occur at any one of the sprockets. Meshing failure leads to a further increase in vibration noise and wear.
Each of the camshaft sprockets A12a and A12b has a number of teeth greater than the number of teeth on the valve-driving crankshaft sprocket A11A, and the number of links engaged with each of these camshaft sprockets is larger than the number of links engaged with the valve driving crankshaft sprocket. The balancer driving crankshaft sprocket A11b has a larger take-up angle than the balancer shaft sprocket A13. In each case, the above-mentioned tight meshing engagement causes at least a part of the chain to travel outside the design chain traveling line, that is, the arc on which the pitch line of the silent chain should travel when properly seated on a sprocket, so that the angular pitch of the chain is equal to the angular pitch of the sprocket. This leads to problems that affect noise properties, vibration properties, friction properties, traveling stability and the like.
Accordingly objects of the invention are to overcome the above-mentioned problems of the prior art, and to provide a silent chain transmission mechanism which allows for. easier assembly of the chain and a reduced requirement for manufacturing accuracy, but in which the silent chain travels in a regular manner over sprockets, on the design chain traveling line, without dislodging. It is also an object of the invention to achieve improved noise, vibration, and friction properties, and travel stability.